Perfect Sorting by Reversals and Deletions/Insertions
|
|
- Mae Poole
- 5 years ago
- Views:
Transcription
1 The Ninth International Symposium on Operations Research and Its Applications (ISORA 10) Chengdu-Jiuzhaigou, China, August 19 23, 2010 Copyright 2010 ORSC & APORC, pp Perfect Sorting by Reversals and Deletions/Insertions Hong-Yu Chen 1 Xiang Tan 2,1 Guo-Jun Li 1, 1 School of Mathematics, Shandong University, Jinan, Shandong, , China 2 School of Statistics and Mathematics, Shandong University of Finance, Jinan, Shandong, , China Abstract Recently, more and more people are interested in the problem of computing a sequence of rearrangement operations to transform one genome into the other such that these operations conserve all common intervals. Bérard et al. show an algorithm to solve the problem in subquadratic time when all operations are reversals. In this paper, we extend the result to allow deletions and a limited form of insertions (which forbids duplications), allowing to compare genomes containing different genes. We provide an exact algorithm for the asymmetric case and a heuristic algorithm for the symmetric case. Keywords algorithm, perfect sorting, common intervals 1 Introduction Genome rearrangement problem is to infer a sequence of rearrangement operations which transform one genome into other genome, minimizing the number of rearrangement operations. In [6], Hannenhalli and Pevzner (HP) developed the first polynomial algorithm for sorting by reversals. El-Mabrouk extended that result to allow deletions and a limited form of insertions (which forbids duplications) in [4]. Recently, another combinatorial framework for sorting by reversals was proposed, called perfect sorting by reversals [5]. It is to infer a minimum number of reversals that don t break any common interval of the considered genome. For this problem, the first algorithm proposed has an exponential worst-case running time [5]. Bérard et al. [1] improved it by showing that the problem is fixed parameterized tractable [3] (FPT): i.e., the complexity function is f (p) n O(1) for some parameter p, where p is the number of prime edges of the strong interval tree of the signed permutation. Recently, they described a more efficient algorithm for the problem in [2]. The worst-case time complexity of their algorithm is parameterized by the maximum prime degree d of the strong interval tree, i.e., f (d) n O(1), where d is always smaller than or equal to the number of prime edges p. In [7], it was showed that when, for a signed permutation, there exists a parsimonious scenario that is also a perfect scenario, computing such a scenario can be done in polynomial time. In this paper, we will extend the result of [1] to include deletions and insertions of gene segments, allowing to compare genomes containing different genes. Corresponding author. address: guojun@csbl.bmb.uga.edu
2 Perfect Sorting by Reversals and Deletions/Insertions 513 Some notation and definitions are introduced in Section 2, then we extend Bérard s method to take into account deletions or insertions in Section 3 and deletions and insertions in Section 4. 2 Preliminaries A signed permutation on n elements is a permutation on the set of integers {1,2,...,n} in which each element has a sign, positive or negative. An interval of a signed permutation is a segment of consecutive elements of the permutation. One can define an interval by giving the set of its unsigned elements, called the content of the interval. The reversal of an interval of a signed permutation reverses the order of the elements of the interval, while changing their signs, that is, a reversal ρ(i, j) rearranges the genes inside the genome and transforms the genome (x 1...x i 1 x i x i+1...x j x j+1...x n ) into the genome (x 1...x i 1 x j... x i+1 x i x j+1...x n ). We denote G ρ as the new genome obtained from genome G as a result of a reversal ρ. If P is a permutation, we denote P the permutation obtained by reversing the complete permutation P. Let G and H be two signed permutations. A scenario between G and H is a sequence of rearrangement events (reversal, deletion, insertion) that transforms G into H, or G into H. The length of such a scenario is the number of rearrangement events it contains. Two distinct intervals I and J commute if their contents trivially intersect, that is, either I J, or J I, or I J = /0. Else, they overlap. We call a reversal ρ(i, j) breaks an interval I = (x a...x b ) if the two intervals (x i...x j ) and (x a...x b ) overlap. If there is a reversal ρ i (1 i t) of a reversal sequence ρ 1,...,ρ t breaks I, we call the reversal sequence ρ 1,...,ρ t breaks I. A common interval of a permutation G and H is an interval in both G and H. The singletons in both G and H are called trivial common intervals. A scenario S between G and H is called a per f ect scenario if every element of S does not break any common interval of G and H. A perfect scenario of minimal length is called a parsimonious per f ect scenario. Let G and H be two genomes with some genes in common, others specific to each genome, and no gene appearing more than once. Write A for the set of genes in both G and H, and A G and A H for those in G and H only respectively. 3 The algorithm 3.1 Background Given two genomes G and H with A G = A H = /0, the problem of sorting by reversals is to find the minimum number of reversals that transform G into H. It is solved by Hannenhalli and Pevzner [6]. They gave an exact polynomial algorithm, denoted by HP algorithm. For two genomes G and H with A G /0,A H = /0, El-Mabrouk [4] gave a new method to represent G and H by the breakpoint graph G (G,H), and presented an algorithm that transforms genome G to H, called Reversal-deletion algorithm. Indeed, it is commonly accepted in computational biology, that if a group of homologous genes (that are genes having a common ancestry) is co-localized in two different
3 514 The 9th International Symposium on Operations Research and Its Applications species, then those genes were probably together in the common ancestor and were not later separated during evolution. Such conserved clusters of homologous genes are called common intervals. A reversal ρ(i, j) is called per f ect if it doesn t break any common interval of G. The problem of per f ect sorting by reversals is to find a reversal sequence ρ 1,...,ρ t such that ρ i (i = 1,...,t) doesn t break any common interval of G, G ρ 1...ρ t = H and t is minimum, where A G = A H = /0. It is solved by Bérard et al. in [1] and [2]. We call their algorithm BBCP algorithm. 3.2 Perfect sorting by reversals and deletions (or insertions) From this section, our goal is to propose algorithms for finding the minimum number of rearrangement operations (reversals, insertions and deletions) necessary to transform genome G into genome H such that they don t break any common interval of G. First, we consider the case when A G /0, A H = /0. It is obvious we only need reversals and deletions to transform G into H, and the genes of A G are destined to be deleted. The following lemma from [5] is fundamental. Lemma 1. If a sequence of reversals sorts a permutation and does not break an interval I, then there exists a sorting sequence of same size (with the same reversals) in which all the reversals contained in I (they sort I) are before all the other reversals (they sort outside I). The following theorem is immediately by Lemma 1. It is the basic idea of our algorithms. Theorem 1. If a sequence of reversals, insertions and deletions sorts a signed permutation and does not break any common interval, then there exists a sorting sequence of same size (with the same reversals, insertions and deletions), in which all the reversals contained in the common intervals are before all the other reversals, insertions and deletions. Proof. It is obvious that all the reversals contained in the common intervals are before all the other reversals by Lemma 1. Since any insertion or deletion doesn t break any common interval, that is, the insertions and deletions are external to all the common intervals, so the insertions and deletions don t overlap with the reversals contained in the common intervals. Hence all the reversals contained in the common intervals could be located before the insertions and deletions. The result follows. Example : Let G = ( ), H = ( ) The common intervals of G and H are {2,3}, {1,2,3}, {4,5} and the singletons {1}, {2}, {3}, {4} and {5}. Reversals {1,2,3,4,5,7}, {1,2,3,6}, {2,3}, {5} and deletion {6,7} is a perfect scenario that transforms G into H. Change the reversals order such that all the reversals contained in the common intervals are before all the other reversals, deletions, that is, the reversals {2, 3}, {5}, {1,2,3,4,5,7}, {1,2,3,6} and deletion {6,7} is also a perfect scenario that transforms G into H. The general idea of our algorithm is as follows: since the sequence of reversals, insertions and deletions can be rearranged such that the reversals contained in the common
4 Perfect Sorting by Reversals and Deletions/Insertions 515 intervals are before all the other operations by Theorem 1, we can sort the common intervals of G first, then transform the obtained permutation G to the other permutation H by El-Mabrouk s method. Let P = {P 1,P 2,...,P k } be a subset of common intervals of G and H, where P i 2, every other common interval is a subset of P i (1 i k) and P i P j = /0 (1 i, j k). Now we give the following algorithm to solve the problem of perfect sorting by reversals and deletions. Algorithm 1: Perfect sorting by reversals and deletions S is an empty scenario. For each common interval P i P, denote S i be the corresponding scenario by applying BBCP algorithm Add S i to S. Denote the obtained permutation after executing the reversals of each S i by G. End for For G and H, applying Reversal-deletion algorithm to transform G to H, denote S be the corresponding scenario Add S to S. End for Theorem 2. Algorithm 1 performs RD(G,H) = R(G,G ) + RD(G,H) reversals and deletions, where R(G,G ) is the total number of reversals that sort each P i of P, and RD(G,H) = R( G,H) + D(G,H) is the reversals and deletions that transform G into H. Moreover, RD(G, H) is the minimum number of rearrangement operations necessary to transform G into H and don t break any common interval. Proof. First, we prove Algorithm 1 is feasible. Since the reversals contained in each common interval can exist before all the other reversals and deletions by Theorem 1, Algorithm 1 sorts each maximal common interval first. In the breakpoint graph G (G,H), the vertices corresponding to the genes of each common interval P i belong to the cycles of size 1 by the definition of breakpoint graph. Since applying Reversal-deletion algorithm doesn t affect the cycles of size 1, then the step of transforming G to H by Reversal-deletion algorithm doesn t break any common interval and the deletions don t affect the common interval obviously. Hence Algorithm 1 is feasible. By BBCP algorithm, R(G,G ) reversals sorting each P i of P is minimum. By the Reversal-deletion algorithm, R( G,H) + D(G,H) is minimum. Hence, RD(G,H) = R(G,G ) + RD(G,H) is the minimum number of rearrangement operations necessary to transform G into H. The result follows. The case A G = /0, A H /0, where the set of genes of G is a subset of the set of genes of H, i.e. the problem of transforming G into H with a minimum number of reversals and insertions such that the operations don t break any common interval, can be solved by Algorithm 1, where G takes on the role of H, and vice versa. Each deletion in the H-to-G solution becomes an insertion in the G-to-H. Each reversal in one direction corresponds to a reversal of the same segment in the opposite direction.
5 516 The 9th International Symposium on Operations Research and Its Applications 3.3 Perfect sorting by reversals, deletions and insertions First, we give the following definition. Definition[4]: the filled genome of G Let G and H be two genomes that A G = /0, G 1 be the graph containing only cycles of size 1 obtained by applying the HP algorithm to the graph G (H,G). Each indirect black edge of G 1 is labeled by a sequence of genes of A H. The genome G c obtained from G as follows: for each pair of adjacent vertices a and b in G, if (a,b) is an indirect black edge of G 1 labeled by the sequence of genes y 1,...,y p of A H, then insert this sequence of genes between a and b in G. We call G c the filled genome of G relative to H. Now, we consider the general case where A G /0 and A H /0. Let G = G \ A G, H = H \ A H, G c be the filled genome of G relative to H, a and b be two adjacent vertices in G, separated in G by a segment X(a) of vertices of A G. If a and b are separated in G c by a segment Y (a) of vertices of A H, then we have the choice to place X(a) before or after Y (a). This choice is made so that we require as few deletions as possible to transform G c into H. By G c we denote the genome obtained by appropriately adding the vertices of A G to G c. The graph G (G c,h) is obtained from G ( G c,h) by transforming certain black edges of G ( G c,h) into indirect edges. El-Mabrouk [4] gave the construction of G c, called Procedure Construct-G c. Now we give the algorithm of perfect sorting by reversals, deletions and insertions. Algorithm 2: Perfect sorting by reversals, deletions and insertions. Step 1. For each common interval P i P, i = 1,...,k, apply BBCP algorithm to sort them respectively. Denote the obtained permutation from G by G. Step 2. Apply the HP algorithm to the graph G (H, G ). The final graph obtained contains D(H, G ) indirect edges. Step 3. Do the D(H, G ) insertions deducible from the indirect edges, and transform G to G c. Step 4. Fill the genome G c to obtain genome G c, using Procedure Construct-G c. Step 5. Apply Reversal-deletion algorithm to the graph G (G c,h). The algorithm does R( G, H) reversals and D(G c,h) deletions. Lemma 2 [4] The number of rearrangement operations that transform G to H during the application of the method is RDI(G,H) = R( G, H) + D(H, G ) + D(G c,h), where R( G, H) is the number of reversals, D(H, G ) is the number of insertions and D(G c,h) is the number of deletions. Moreover, if D(G c,h) = D(G, H), then the total number of rearrangement operations is minimal. Theorem 3. Algorithm 2 performs RDI(G,H) = R(G,G ) + RDI(G,H) reversals, deletions and insertions, where R(G,G ) is the total number of reversals that sort each P i of P, and RDI(G,H) = R( G, H) + D(H, G ) + D(G c,h) is the reversals, deletions and insertions that transform G into H. Moreover, the rearrangement operations necessary to
6 Perfect Sorting by Reversals and Deletions/Insertions 517 transform G into H don t break any common interval and if D(G c,h) = D(G, H), then it is minimal. Proof. By Theorem 1, we can sort each common interval first, then transform the obtained genome to H. After Step 1, in the graph G (H, G ) and G (G c,h), the vertices corresponding to the genes of each common interval P i belong to the cycles of size 1. With a similar proof to that of Theorem 2, Step 2 to Step 5 doesn t break any common interval. So Algorithm 2 is feasible. It is obvious Algorithm 2 performs RDI(G,H) = R(G,G ) + R( G, H) + D(H, G ) + D(G c,h) by the five steps. By BBCP algorithm, R(G,G ) reversals that transform G to G is minimal. By Lemma 2, if D(G c,h) = D(G, H), then the total number of rearrangement operations R( G, H)+ D(H, G ) + D(G c,h) that transform G to H is minimal. So if D(G c,h) = D(G, H), then the rearrangement operations necessary to transform G into H is minimal. The result follows. Lemma 3([1]) Assume m = max( P i i = 1,...,k). The transformation of G to G by BBCP algorithm is of time complexity O(m mlogm) if each T S (P i ) is unambiguous; O(2 p m mlogm), if there is some ambiguous T S (P i ), where p is the maximum number of unsigned prime vertices of T S (P i ). Lemma 4([4]) Assume n = max( A + A G, A + A H ). The transformation of G to H by El-Mabrouk s method is of time complexity O(n 2 ). It is obvious that n > m. By Lemma 3 and Lemma 4, the following theorem is immediately. Theorem 4. The transformation of G to H by Algorithm 2 is of time complexity O(n 2 ), when each T S (P i ) is unambiguous; O(2 p m mlogm + n 2 ), when there is some ambiguous T S (P i ), where p is the maximum number of unsigned prime vertices of T S (P i ). 4 Conclusion Our main result in this note is an algorithm that finds the minimum rearrangement operations necessary to transform genome G into H, where G and H have different genes, such that these operations don t break any common interval. Obviously, in such case, insertions or deletions are needed. We combine a strong interval tree with a breakpoint graph to show how to extend the BBCP algorithm to an algorithm that allows deletions and a limited form of insertions. The next step of this work will be to generalize our approach to handle duplications as well as insertions and present an algorithm for computing (near) minimal edit sequences involving insertions, deletions and reversals. Acknowledgements This work was supported by National Natural Science Foundation of China ( , , ), and also partially supported by Graduate Independent Innovation Foundation of Shandong University ( ).
7 518 The 9th International Symposium on Operations Research and Its Applications References [1] S. Bérard, A. Bergeron, C. Chauve, C.Paul, Perfect sorting by reversals is not always difficult, IEEE/ACM Trans. Comput. Biology Bioinform, 4(1) (2007) [2] S. Bérard, C. Chauve, C.Paul, A more efficient algorithm for perfect sorting by reversals, Information Processing Letters, 106 (2008) [3] R.G. Downey, M.R. Fellows, Parameterized Complexity, springer, [4] N. El-Mabrouk, Genome rearrangement by reversals and insertions/deletions of contiguous segments, In proceeding 11th Ann. Symp. Combin. Pattern Matching CPM 00, London: springer, 1848 (2000) [5] M. Figeac, J.-S Varré, Sorting by reversals with common intervals, Proceedings of WABI 2004, Lecture Notes in Computer Sciences, Berlin: springer, 3240 (2004) [6] S. Hannenhalli, P.A. Pevzner, Transforming cabbage into turnip (Polynomial algorithm for sorting signed permutations by reversals) In proceedings of the 27th Annual ACM-SIAM Symposium on the Theory of Computing, (1995) [7] M. -F. Sagot, E. Tannier, Perfect sorting by reversals, Proceedings of COCOON 2005, Lecture Notes in Computer Sciences, Berlin: springer, 3595 (2005)
On the complexity of unsigned translocation distance
Theoretical Computer Science 352 (2006) 322 328 Note On the complexity of unsigned translocation distance Daming Zhu a, Lusheng Wang b, a School of Computer Science and Technology, Shandong University,
More informationThe combinatorics and algorithmics of genomic rearrangements have been the subject of much
JOURNAL OF COMPUTATIONAL BIOLOGY Volume 22, Number 5, 2015 # Mary Ann Liebert, Inc. Pp. 425 435 DOI: 10.1089/cmb.2014.0096 An Exact Algorithm to Compute the Double-Cutand-Join Distance for Genomes with
More informationAN EXACT SOLVER FOR THE DCJ MEDIAN PROBLEM
AN EXACT SOLVER FOR THE DCJ MEDIAN PROBLEM MENG ZHANG College of Computer Science and Technology, Jilin University, China Email: zhangmeng@jlueducn WILLIAM ARNDT AND JIJUN TANG Dept of Computer Science
More informationGenome Rearrangements In Man and Mouse. Abhinav Tiwari Department of Bioengineering
Genome Rearrangements In Man and Mouse Abhinav Tiwari Department of Bioengineering Genome Rearrangement Scrambling of the order of the genome during evolution Operations on chromosomes Reversal Translocation
More informationA Framework for Orthology Assignment from Gene Rearrangement Data
A Framework for Orthology Assignment from Gene Rearrangement Data Krister M. Swenson, Nicholas D. Pattengale, and B.M.E. Moret Department of Computer Science University of New Mexico Albuquerque, NM 87131,
More informationAlgorithms for Bioinformatics
Adapted from slides by Alexandru Tomescu, Leena Salmela, Veli Mäkinen, Esa Pitkänen 582670 Algorithms for Bioinformatics Lecture 5: Combinatorial Algorithms and Genomic Rearrangements 1.10.2015 Background
More informationScaffold Filling Under the Breakpoint Distance
Scaffold Filling Under the Breakpoint Distance Haitao Jiang 1,2, Chunfang Zheng 3, David Sankoff 4, and Binhai Zhu 1 1 Department of Computer Science, Montana State University, Bozeman, MT 59717-3880,
More informationSorting Unsigned Permutations by Weighted Reversals, Transpositions, and Transreversals
Lou XW, Zhu DM. Sorting unsigned permutations by weighted reversals, transpositions, and transreversals. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 25(4): 853 863 July 2010. DOI 10.1007/s11390-010-1066-7
More informationGene Maps Linearization using Genomic Rearrangement Distances
Gene Maps Linearization using Genomic Rearrangement Distances Guillaume Blin Eric Blais Danny Hermelin Pierre Guillon Mathieu Blanchette Nadia El-Mabrouk Abstract A preliminary step to most comparative
More informationPhysical Mapping. Restriction Mapping. Lecture 12. A physical map of a DNA tells the location of precisely defined sequences along the molecule.
Computat onal iology Lecture 12 Physical Mapping physical map of a DN tells the location of precisely defined sequences along the molecule. Restriction mapping: mapping of restriction sites of a cutting
More informationarxiv: v1 [cs.ds] 21 May 2013
Easy identification of generalized common nested intervals Fabien de Montgolfier 1, Mathieu Raffinot 1, and Irena Rusu 2 arxiv:1305.4747v1 [cs.ds] 21 May 2013 1 LIAFA, Univ. Paris Diderot - Paris 7, 75205
More informationPhylogenetic Networks, Trees, and Clusters
Phylogenetic Networks, Trees, and Clusters Luay Nakhleh 1 and Li-San Wang 2 1 Department of Computer Science Rice University Houston, TX 77005, USA nakhleh@cs.rice.edu 2 Department of Biology University
More informationThe Complexity of Maximum. Matroid-Greedoid Intersection and. Weighted Greedoid Maximization
Department of Computer Science Series of Publications C Report C-2004-2 The Complexity of Maximum Matroid-Greedoid Intersection and Weighted Greedoid Maximization Taneli Mielikäinen Esko Ukkonen University
More informationA 3-APPROXIMATION ALGORITHM FOR THE SUBTREE DISTANCE BETWEEN PHYLOGENIES. 1. Introduction
A 3-APPROXIMATION ALGORITHM FOR THE SUBTREE DISTANCE BETWEEN PHYLOGENIES MAGNUS BORDEWICH 1, CATHERINE MCCARTIN 2, AND CHARLES SEMPLE 3 Abstract. In this paper, we give a (polynomial-time) 3-approximation
More informationMinimum Common String Partition Problem: Hardness and Approximations
Minimum Common String Partition Problem: Hardness and Approximations Avraham Goldstein Department of Mathematics Stony Brook University Stony Brook, NY, U.S.A. avi goldstein@netzero.net Petr Kolman Department
More informationAnalysis of Gene Order Evolution beyond Single-Copy Genes
Analysis of Gene Order Evolution beyond Single-Copy Genes Nadia El-Mabrouk Département d Informatique et de Recherche Opérationnelle Université de Montréal mabrouk@iro.umontreal.ca David Sankoff Department
More informationThe Intractability of Computing the Hamming Distance
The Intractability of Computing the Hamming Distance Bodo Manthey and Rüdiger Reischuk Universität zu Lübeck, Institut für Theoretische Informatik Wallstraße 40, 23560 Lübeck, Germany manthey/reischuk@tcs.uni-luebeck.de
More informationReversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set
Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set Petr Kolman Charles University in Prague Faculty of Mathematics and Physics Department of Applied Mathematics
More informationGreedy Algorithms. CS 498 SS Saurabh Sinha
Greedy Algorithms CS 498 SS Saurabh Sinha Chapter 5.5 A greedy approach to the motif finding problem Given t sequences of length n each, to find a profile matrix of length l. Enumerative approach O(l n
More informationGene Maps Linearization using Genomic Rearrangement Distances
Gene Maps Linearization using Genomic Rearrangement Distances Guillaume Blin, Eric Blais, Danny Hermelin, Pierre Guillon, Mathieu Blanchette, Nadia El-Mabrouk To cite this version: Guillaume Blin, Eric
More informationA PARSIMONY APPROACH TO ANALYSIS OF HUMAN SEGMENTAL DUPLICATIONS
A PARSIMONY APPROACH TO ANALYSIS OF HUMAN SEGMENTAL DUPLICATIONS CRYSTAL L. KAHN and BENJAMIN J. RAPHAEL Box 1910, Brown University Department of Computer Science & Center for Computational Molecular Biology
More informationClassical Complexity and Fixed-Parameter Tractability of Simultaneous Consecutive Ones Submatrix & Editing Problems
Classical Complexity and Fixed-Parameter Tractability of Simultaneous Consecutive Ones Submatrix & Editing Problems Rani M. R, Mohith Jagalmohanan, R. Subashini Binary matrices having simultaneous consecutive
More informationAn experimental investigation on the pancake problem
An experimental investigation on the pancake problem Bruno Bouzy Paris Descartes University IJCAI Computer Game Workshop Buenos-Aires July, 0 Outline The pancake problem The genome rearrangement problem
More informationInferring positional homologs with common intervals of sequences
Outline Introduction Our approach Results Conclusion Inferring positional homologs with common intervals of sequences Guillaume Blin, Annie Chateau, Cedric Chauve, Yannick Gingras CGL - Université du Québec
More informationDiscrete Applied Mathematics
Discrete Applied Mathematics 159 (2011) 1641 1645 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note Girth of pancake graphs Phillip
More informationApproximation Algorithms for the Consecutive Ones Submatrix Problem on Sparse Matrices
Approximation Algorithms for the Consecutive Ones Submatrix Problem on Sparse Matrices Jinsong Tan and Louxin Zhang Department of Mathematics, National University of Singaproe 2 Science Drive 2, Singapore
More informationarxiv: v2 [cs.dm] 12 Jul 2014
Interval Scheduling and Colorful Independent Sets arxiv:1402.0851v2 [cs.dm] 12 Jul 2014 René van Bevern 1, Matthias Mnich 2, Rolf Niedermeier 1, and Mathias Weller 3 1 Institut für Softwaretechnik und
More informationMultiple Whole Genome Alignment
Multiple Whole Genome Alignment BMI/CS 776 www.biostat.wisc.edu/bmi776/ Spring 206 Anthony Gitter gitter@biostat.wisc.edu These slides, excluding third-party material, are licensed under CC BY-NC 4.0 by
More informationA Fixed-Parameter Algorithm for Max Edge Domination
A Fixed-Parameter Algorithm for Max Edge Domination Tesshu Hanaka and Hirotaka Ono Department of Economic Engineering, Kyushu University, Fukuoka 812-8581, Japan ono@cscekyushu-uacjp Abstract In a graph,
More informationSorting by Transpositions is Difficult
Sorting by Transpositions is Difficult Laurent Bulteau, Guillaume Fertin, Irena Rusu Laboratoire d Informatique de Nantes-Atlantique (LINA), UMR CNRS 6241 Université de Nantes, 2 rue de la Houssinière,
More informationCS781 Lecture 3 January 27, 2011
CS781 Lecture 3 January 7, 011 Greedy Algorithms Topics: Interval Scheduling and Partitioning Dijkstra s Shortest Path Algorithm Minimum Spanning Trees Single-Link k-clustering Interval Scheduling Interval
More informationExact Algorithms for Dominating Induced Matching Based on Graph Partition
Exact Algorithms for Dominating Induced Matching Based on Graph Partition Mingyu Xiao School of Computer Science and Engineering University of Electronic Science and Technology of China Chengdu 611731,
More informationA Cubic-Vertex Kernel for Flip Consensus Tree
To appear in Algorithmica A Cubic-Vertex Kernel for Flip Consensus Tree Christian Komusiewicz Johannes Uhlmann Received: date / Accepted: date Abstract Given a bipartite graph G = (V c, V t, E) and a nonnegative
More informationParameterized Algorithms and Kernels for 3-Hitting Set with Parity Constraints
Parameterized Algorithms and Kernels for 3-Hitting Set with Parity Constraints Vikram Kamat 1 and Neeldhara Misra 2 1 University of Warsaw vkamat@mimuw.edu.pl 2 Indian Institute of Science, Bangalore neeldhara@csa.iisc.ernet.in
More informationOn Machine Dependency in Shop Scheduling
On Machine Dependency in Shop Scheduling Evgeny Shchepin Nodari Vakhania Abstract One of the main restrictions in scheduling problems are the machine (resource) restrictions: each machine can perform at
More informationParameterized Algorithms for the H-Packing with t-overlap Problem
Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 18, no. 4, pp. 515 538 (2014) DOI: 10.7155/jgaa.00335 Parameterized Algorithms for the H-Packing with t-overlap Problem Alejandro López-Ortiz
More informationThe Budgeted Unique Coverage Problem and Color-Coding (Extended Abstract)
The Budgeted Unique Coverage Problem and Color-Coding (Extended Abstract) Neeldhara Misra 1, Venkatesh Raman 1, Saket Saurabh 2, and Somnath Sikdar 1 1 The Institute of Mathematical Sciences, Chennai,
More informationEdit Distance with Move Operations
Edit Distance with Move Operations Dana Shapira and James A. Storer Computer Science Department shapird/storer@cs.brandeis.edu Brandeis University, Waltham, MA 02254 Abstract. The traditional edit-distance
More informationGraphs, permutations and sets in genome rearrangement
ntroduction Graphs, permutations and sets in genome rearrangement 1 alabarre@ulb.ac.be Universite Libre de Bruxelles February 6, 2006 Computers in Scientic Discovery 1 Funded by the \Fonds pour la Formation
More informationAncestral Genome Organization: an Alignment Approach
Ancestral Genome Organization: an Alignment Approach Patrick Holloway 1, Krister Swenson 2, David Ardell 3, and Nadia El-Mabrouk 4 1 Département d Informatique et de Recherche Opérationnelle (DIRO), Université
More informationarxiv: v1 [cs.ds] 2 Dec 2009
Variants of Constrained Longest Common Subsequence arxiv:0912.0368v1 [cs.ds] 2 Dec 2009 Paola Bonizzoni Gianluca Della Vedova Riccardo Dondi Yuri Pirola Abstract In this work, we consider a variant of
More informationHybrid Fault diagnosis capability analysis of Hypercubes under the PMC model and MM model
Hybrid Fault diagnosis capability analysis of Hypercubes under the PMC model and MM model Qiang Zhu,Lili Li, Sanyang Liu, Xing Zhang arxiv:1709.05588v1 [cs.dc] 17 Sep 017 Abstract System level diagnosis
More informationConsecutive ones Block for Symmetric Matrices
Consecutive ones Block for Symmetric Matrices Rui Wang, FrancisCM Lau Department of Computer Science and Information Systems The University of Hong Kong, Hong Kong, PR China Abstract We show that a cubic
More informationLinear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs
Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs Nathan Lindzey, Ross M. McConnell Colorado State University, Fort Collins CO 80521, USA Abstract. Tucker characterized
More informationAlgebraic Properties and Panconnectivity of Folded Hypercubes
Algebraic Properties and Panconnectivity of Folded Hypercubes Meijie Ma a Jun-Ming Xu b a School of Mathematics and System Science, Shandong University Jinan, 50100, China b Department of Mathematics,
More informationRandomized Algorithms. Lecture 4. Lecturer: Moni Naor Scribe by: Tamar Zondiner & Omer Tamuz Updated: November 25, 2010
Randomized Algorithms Lecture 4 Lecturer: Moni Naor Scribe by: Tamar Zondiner & Omer Tamuz Updated: November 25, 2010 1 Pairwise independent hash functions In the previous lecture we encountered two families
More informationSome hard families of parameterised counting problems
Some hard families of parameterised counting problems Mark Jerrum and Kitty Meeks School of Mathematical Sciences, Queen Mary University of London {m.jerrum,k.meeks}@qmul.ac.uk September 2014 Abstract
More informationExploring Phylogenetic Relationships in Drosophila with Ciliate Operations
Exploring Phylogenetic Relationships in Drosophila with Ciliate Operations Jacob Herlin, Anna Nelson, and Dr. Marion Scheepers Department of Mathematical Sciences, University of Northern Colorado, Department
More information1a. Introduction COMP6741: Parameterized and Exact Computation
1a. Introduction COMP6741: Parameterized and Exact Computation Serge Gaspers 12 1 School of Computer Science and Engineering, UNSW Sydney, Asutralia 2 Decision Sciences Group, Data61, CSIRO, Australia
More informationSolution of Large-scale LP Problems Using MIP Solvers: Repeated Assignment Problem
The Eighth International Symposium on Operations Research and Its Applications (ISORA 09) Zhangjiajie, China, September 20 22, 2009 Copyright 2009 ORSC & APORC, pp. 190 197 Solution of Large-scale LP Problems
More informationOn the hardness of losing width
On the hardness of losing width Marek Cygan 1, Daniel Lokshtanov 2, Marcin Pilipczuk 1, Micha l Pilipczuk 1, and Saket Saurabh 3 1 Institute of Informatics, University of Warsaw, Poland {cygan@,malcin@,mp248287@students}mimuwedupl
More informationTight Approximation Ratio of a General Greedy Splitting Algorithm for the Minimum k-way Cut Problem
Algorithmica (2011 59: 510 520 DOI 10.1007/s00453-009-9316-1 Tight Approximation Ratio of a General Greedy Splitting Algorithm for the Minimum k-way Cut Problem Mingyu Xiao Leizhen Cai Andrew Chi-Chih
More informationCovering Linear Orders with Posets
Covering Linear Orders with Posets Proceso L. Fernandez, Lenwood S. Heath, Naren Ramakrishnan, and John Paul C. Vergara Department of Information Systems and Computer Science, Ateneo de Manila University,
More informationGenome Rearrangement Problems with Single and Multiple Gene Copies: A Review
Genome Rearrangement Problems with Single and Multiple Gene Copies: A Review Ron Zeira and Ron Shamir August 9, 2018 Dedicated to Bernard Moret upon his retirement. Abstract Genome rearrangement problems
More informationSUB-EXPONENTIALLY MANY 3-COLORINGS OF TRIANGLE-FREE PLANAR GRAPHS
SUB-EXPONENTIALLY MANY 3-COLORINGS OF TRIANGLE-FREE PLANAR GRAPHS Arash Asadi Luke Postle 1 Robin Thomas 2 School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332-0160, USA ABSTRACT
More informationOn Reversal and Transposition Medians
On Reversal and Transposition Medians Martin Bader International Science Index, Computer and Information Engineering waset.org/publication/7246 Abstract During the last years, the genomes of more and more
More informationApproximate Map Labeling. is in (n log n) Frank Wagner* B 93{18. December Abstract
SERIE B INFORMATIK Approximate Map Labeling is in (n log n) Frank Wagner* B 93{18 December 1993 Abstract Given n real numbers, the -CLOSENESS problem consists in deciding whether any two of them are within
More informationarxiv: v1 [cs.dm] 29 Oct 2012
arxiv:1210.7684v1 [cs.dm] 29 Oct 2012 Square-Root Finding Problem In Graphs, A Complete Dichotomy Theorem. Babak Farzad 1 and Majid Karimi 2 Department of Mathematics Brock University, St. Catharines,
More informationOn Resolution Limit of the Modularity in Community Detection
The Ninth International Symposium on Operations Research and Its Applications (ISORA 10) Chengdu-Jiuzhaigou, China, August 19 23, 2010 Copyright 2010 ORSC & APORC, pp. 492 499 On Resolution imit of the
More informationReversing Gene Erosion Reconstructing Ancestral Bacterial Genomes from Gene-Content and Order Data
Reversing Gene Erosion Reconstructing Ancestral Bacterial Genomes from Gene-Content and Order Data Joel V. Earnest-DeYoung 1, Emmanuelle Lerat 2, and Bernard M.E. Moret 1,3 Abstract In the last few years,
More informationEvolution of Tandemly Arrayed Genes in Multiple Species
Evolution of Tandemly Arrayed Genes in Multiple Species Mathieu Lajoie 1, Denis Bertrand 1, and Nadia El-Mabrouk 1 DIRO - Université de Montréal - H3C 3J7 - Canada {bertrden,lajoimat,mabrouk}@iro.umontreal.ca
More informationBurnt Pancake Problem : New Results (New Lower Bounds on the Diameter and New Experimental Optimality Ratios)
Burnt Pancake Problem : New Results (New Lower Bounds on the Diameter and New Experimental Optimality Ratios) Bruno Bouzy Paris Descartes University SoCS 06 Tarrytown July 6-8, 06 Outline The Burnt Pancake
More informationGenome Rearrangement Problems with Single and Multiple Gene Copies: A Review
Genome Rearrangement Problems with Single and Multiple Gene Copies: A Review Ron Zeira and Ron Shamir June 27, 2018 Dedicated to Bernard Moret upon his retirement. Abstract Problems of genome rearrangement
More informationFast Algebraic Immunity of 2 m + 2 & 2 m + 3 variables Majority Function
Fast Algebraic Immunity of 2 m + 2 & 2 m + 3 variables Majority Function Yindong Chen a,, Fei Guo a, Liu Zhang a a College of Engineering, Shantou University, Shantou 515063, China Abstract Boolean functions
More informationStrongly chordal and chordal bipartite graphs are sandwich monotone
Strongly chordal and chordal bipartite graphs are sandwich monotone Pinar Heggernes Federico Mancini Charis Papadopoulos R. Sritharan Abstract A graph class is sandwich monotone if, for every pair of its
More informationI519 Introduction to Bioinformatics, Genome Comparison. Yuzhen Ye School of Informatics & Computing, IUB
I519 Introduction to Bioinformatics, 2015 Genome Comparison Yuzhen Ye (yye@indiana.edu) School of Informatics & Computing, IUB Whole genome comparison/alignment Build better phylogenies Identify polymorphism
More informationGALOIS THEORY: LECTURE 20
GALOIS THEORY: LECTURE 0 LEO GOLDMAKHER. REVIEW: THE FUNDAMENTAL LEMMA We begin today s lecture by recalling the Fundamental Lemma introduced at the end of Lecture 9. This will come up in several places
More informationApproximability and Parameterized Complexity of Consecutive Ones Submatrix Problems
Proc. 4th TAMC, 27 Approximability and Parameterized Complexity of Consecutive Ones Submatrix Problems Michael Dom, Jiong Guo, and Rolf Niedermeier Institut für Informatik, Friedrich-Schiller-Universität
More informationIncreasing the Span of Stars
Increasing the Span of Stars Ning Chen Roee Engelberg C. Thach Nguyen Prasad Raghavendra Atri Rudra Gynanit Singh Department of Computer Science and Engineering, University of Washington, Seattle, WA.
More informationBounds on parameters of minimally non-linear patterns
Bounds on parameters of minimally non-linear patterns P.A. CrowdMath Department of Mathematics Massachusetts Institute of Technology Massachusetts, U.S.A. crowdmath@artofproblemsolving.com Submitted: Jan
More informationChromosomal Breakpoint Reuse in Genome Sequence Rearrangement ABSTRACT
JOURNAL OF COMPUTATIONAL BIOLOGY Volume 12, Number 6, 2005 Mary Ann Liebert, Inc. Pp. 812 821 Chromosomal Breakpoint Reuse in Genome Sequence Rearrangement DAVID SANKOFF 1 and PHIL TRINH 2 ABSTRACT In
More informationarxiv: v2 [cs.ds] 2 Dec 2013
arxiv:1305.4747v2 [cs.ds] 2 Dec 2013 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 Easy identification of generalized common and conserved nested intervals Fabien de Montgolfier
More informationDistributionally Robust Discrete Optimization with Entropic Value-at-Risk
Distributionally Robust Discrete Optimization with Entropic Value-at-Risk Daniel Zhuoyu Long Department of SEEM, The Chinese University of Hong Kong, zylong@se.cuhk.edu.hk Jin Qi NUS Business School, National
More informationP,NP, NP-Hard and NP-Complete
P,NP, NP-Hard and NP-Complete We can categorize the problem space into two parts Solvable Problems Unsolvable problems 7/11/2011 1 Halting Problem Given a description of a program and a finite input, decide
More informationAverage Case Analysis. October 11, 2011
Average Case Analysis October 11, 2011 Worst-case analysis Worst-case analysis gives an upper bound for the running time of a single execution of an algorithm with a worst-case input and worst-case random
More informationAhlswede Khachatrian Theorems: Weighted, Infinite, and Hamming
Ahlswede Khachatrian Theorems: Weighted, Infinite, and Hamming Yuval Filmus April 4, 2017 Abstract The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of
More informationCS 350 Algorithms and Complexity
CS 350 Algorithms and Complexity Winter 2019 Lecture 15: Limitations of Algorithmic Power Introduction to complexity theory Andrew P. Black Department of Computer Science Portland State University Lower
More informationSome Algorithmic Challenges in Genome-Wide Ortholog Assignment
Jiang T. Some algorithmic challenges in genome-wide ortholog assignment. JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 25(1): 1 Jan. 2010 Some Algorithmic Challenges in Genome-Wide Ortholog Assignment Tao
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 6.85J / 8.5J Advanced Algorithms Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.5/6.85 Advanced Algorithms
More informationCS 350 Algorithms and Complexity
1 CS 350 Algorithms and Complexity Fall 2015 Lecture 15: Limitations of Algorithmic Power Introduction to complexity theory Andrew P. Black Department of Computer Science Portland State University Lower
More informationChain Minors are FPT. B Marcin Kamiński. 1 Introduction. Jarosław Błasiok 1 Marcin Kamiński 1
Algorithmica (2017) 79:698 707 DOI 10.1007/s00453-016-0220-1 Chain Minors are FPT Jarosław Błasiok 1 Marcin Kamiński 1 Received: 21 March 2014 / Accepted: 21 September 2016 / Published online: 20 October
More informationCh6 Addition Proofs of Theorems on permutations.
Ch6 Addition Proofs of Theorems on permutations. Definition Two permutations ρ and π of a set A are disjoint if every element moved by ρ is fixed by π and every element moved by π is fixed by ρ. (In other
More informationDominating Set Counting in Graph Classes
Dominating Set Counting in Graph Classes Shuji Kijima 1, Yoshio Okamoto 2, and Takeaki Uno 3 1 Graduate School of Information Science and Electrical Engineering, Kyushu University, Japan kijima@inf.kyushu-u.ac.jp
More informationLecture 6. Today we shall use graph entropy to improve the obvious lower bound on good hash functions.
CSE533: Information Theory in Computer Science September 8, 010 Lecturer: Anup Rao Lecture 6 Scribe: Lukas Svec 1 A lower bound for perfect hash functions Today we shall use graph entropy to improve the
More informationarxiv: v1 [cs.ds] 22 Apr 2013
Chain minors are FPT Jaros law B lasiok 1 and Marcin Kamiński 2 1 Instytut Informatyki Uniwersytet Warszawski jb291202@students.mimuw.edu.pl arxiv:1304.5849v1 [cs.ds] 22 Apr 2013 2 Département d Informatique
More informationMultiple genome rearrangement
Multiple genome rearrangement David Sankoff Mathieu Blanchette 1 Introduction Multiple alignment of macromolecular sequences, an important topic of algorithmic research for at least 25 years [13, 10],
More informationLet S be a set of n species. A phylogeny is a rooted tree with n leaves, each of which is uniquely
JOURNAL OF COMPUTATIONAL BIOLOGY Volume 8, Number 1, 2001 Mary Ann Liebert, Inc. Pp. 69 78 Perfect Phylogenetic Networks with Recombination LUSHENG WANG, 1 KAIZHONG ZHANG, 2 and LOUXIN ZHANG 3 ABSTRACT
More informationThe Greedy Algorithm for the Symmetric TSP
The Greedy Algorithm for the Symmetric TSP Gregory Gutin Anders Yeo Abstract We corrected proofs of two results on the greedy algorithm for the Symmetric TSP and answered a question in Gutin and Yeo, Oper.
More informationCorrelation of WNCP Curriculum to Pearson Foundations and Pre-calculus Mathematics 10
Measurement General Outcome: Develop spatial sense and proportional reasoning. 1. Solve problems that involve linear measurement, using: SI and imperial units of measure estimation strategies measurement
More informationThe Generalized Regenerator Location Problem
The Generalized Regenerator Location Problem Si Chen Ivana Ljubić S. Raghavan College of Business and Public Affairs, Murray State University Murray, KY 42071, USA Faculty of Business, Economics, and Statistics,
More informationAn approximation algorithm for the partial covering 0 1 integer program
An approximation algorithm for the partial covering 0 1 integer program Yotaro Takazawa Shinji Mizuno Tomonari Kitahara submitted January, 2016 revised July, 2017 Abstract The partial covering 0 1 integer
More informationAnalogies and discrepancies between the vertex cover number and the weakly connected domination number of a graph
Analogies and discrepancies between the vertex cover number and the weakly connected domination number of a graph M. Lemańska a, J. A. Rodríguez-Velázquez b, Rolando Trujillo-Rasua c, a Department of Technical
More informationTRIPLE FACTORIZATION OF NON-ABELIAN GROUPS BY TWO MAXIMAL SUBGROUPS
Journal of Algebra and Related Topics Vol. 2, No 2, (2014), pp 1-9 TRIPLE FACTORIZATION OF NON-ABELIAN GROUPS BY TWO MAXIMAL SUBGROUPS A. GHARIBKHAJEH AND H. DOOSTIE Abstract. The triple factorization
More informationFactorization of integer-valued polynomials with square-free denominator
accepted by Comm. Algebra (2013) Factorization of integer-valued polynomials with square-free denominator Giulio Peruginelli September 9, 2013 Dedicated to Marco Fontana on the occasion of his 65th birthday
More informationEfficient Polynomial-Time Algorithms for Variants of the Multiple Constrained LCS Problem
Efficient Polynomial-Time Algorithms for Variants of the Multiple Constrained LCS Problem Hsing-Yen Ann National Center for High-Performance Computing Tainan 74147, Taiwan Chang-Biau Yang and Chiou-Ting
More informationCardinality Networks: a Theoretical and Empirical Study
Constraints manuscript No. (will be inserted by the editor) Cardinality Networks: a Theoretical and Empirical Study Roberto Asín, Robert Nieuwenhuis, Albert Oliveras, Enric Rodríguez-Carbonell Received:
More informationTHE KNUTH RELATIONS HUAN VO
THE KNUTH RELATIONS HUAN VO Abstract We define three equivalence relations on the set of words using different means It turns out that they are the same relation Motivation Recall the algorithm of row
More informationFRACTIONAL PACKING OF T-JOINS. 1. Introduction
FRACTIONAL PACKING OF T-JOINS FRANCISCO BARAHONA Abstract Given a graph with nonnegative capacities on its edges, it is well known that the capacity of a minimum T -cut is equal to the value of a maximum
More informationSTABLE MARRIAGE PROBLEM WITH TIES AND INCOMPLETE BOUNDED LENGTH PREFERENCE LIST UNDER SOCIAL STABILITY
STABLE MARRIAGE PROBLEM WITH TIES AND INCOMPLETE BOUNDED LENGTH PREFERENCE LIST UNDER SOCIAL STABILITY Ashish Shrivastava and C. Pandu Rangan Department of Computer Science and Engineering, Indian Institute
More information